What Is the Identity Function in Math?

The identity function in math is one in which the output of the function is equal to its input, often written as f(x) = x for all x. The input-output pair made up of x and y are always identical, thus the name identity function. This holds true not only for the set of all real numbers, but also for the set of all real functions. Often considered mathematically trivial, the identity function is the basis for all other functions.

Casting the identity function as a linear function of the form y = mx + b, where m = 1 and b = 0 for all real numbers, more properties of the identity function become easier to see. All linear functions are combinations of the identity function and two constant functions. For example, the linear function y = 3x + 2 breaks down into the identity function multiplied by the constant function y = 3, then added to the constant function y = 2. Conversely, the identity function is a special case of all linear functions.

Although there are not many practical real-world applications of the identity function, it is also true that the identify function underlies all practical real-world applications. It is most frequently used in the theoretical or abstract fields of mathematics, each of which may or may not have direct real-world applications. It is occasionally useful in computer science applications when a function requires that its arguments be functions. In certain cases then, the identity function f(x) can replace the simple variable x.